On tail behaviour of $k$-th upper order statistics under fixed and random sample sizes via tail equivalence

TL;DR

This paper investigates the tail behaviors of limit laws of normalized $k$-th upper order statistics under fixed and random sample sizes, revealing their max domains of attraction and stochastic orderings through tail equivalence.

Contribution

It provides a comprehensive analysis of tail behaviors of $k$-th upper order statistics' limit laws, generalizing existing results and applying to power norming with elementary proofs.

Findings

Identifies tail behaviors of limit laws under various sample size conditions.

Determines max domains of attraction for the limit laws.

Establishes stochastic ordering properties of the limit laws.

Abstract

For a fixed positive integer $\;k,\;$ limit laws of linearly normalized $\;k$-th upper order statistics are well known. In this article, a comprehensive study of tail behaviours of limit laws of normalized $k$-th upper order statistics under fixed and random sample sizes is carried out using tail equivalence which leads to some interesting tail behaviours of the limit laws. These lead to definitive answers about their max domains of attraction. Stochastic ordering properties of the limit laws are also studied. The results obtained are not dependent on linear norming and apply to power norming also and generalize some results already available in the literature. And the proofs given here are elementary.